Decomposing tensor products of irreducible representations of reductive groups over a finite field

Question

Let $G$ be a reductive group over a finite field (i.e. finite groups over lie type). The case I am most interested in is $G=GL_{n}(\mathbb{F}_{q})$; other classical groups are also interesting I think.

Deligne-Lusztig theory has a lot to say about the irreducible representations and characters of these groups. For $G=GL_n(\mathbb{F}_q)$, Green's paper from the 1940's gives the characters explicitly also. The following question I guess, is in part a reference request, since the question has probably been examined in the literature somewhere, but I am unable to find a reference.

Question: Let $V$ and $W$ be two irreducible representations of $G$. What can be said about the decomposition of $V \otimes W$ into irreducibles? Specifically:

• Are there any special cases of $V$ for which the decomposition of $V \otimes W$ into irreducibles can always be explicitly determined? (for instance, with the symmetric group $S_n$, there is some theory which does this for the regular representation of dimension $n-1$, and also I believe work which does this for representations corresponding to two-row partitions).
• Is there anything that can be said for the decomposition of $V \otimes V$ in general?
• What about, if $V$ and $W$ are not actually irreducibles, but instead representations obtained from $l$-adic cohomology; for instance the virtual representation $R_{T, \theta}$ is defined as alternating sums of various cohomological representations. As an example, consider the representation of $G$ acting on the $i$-th cohomology of the Deligne-Lusztig variety $X_{T}$ corresponding to a fixed torus $T$; if we tensor together two different cohomological representations corresponding to different tori, and cohomology for different values of $i$, what can we say? Since the $R_{T, \theta}$ are defined as alternating sums of these, perhaps this question will help with our original problem.

Answer 1

Theorem 1.4.1 in arxiv:0810.2076 answers some of your questions for generic semisimple irreducible representations. Emmanuel Letellier has hitherto unpublished results where he does answer your question for all generic irreducible representations in terms of intersection cohomology of certain quiver varieties. We did not know about other results on the representation ring of $GL_n({\mathbb F}_q)$. EDIT: but see Victor's answer for related results of Lusztig. EDIT 2 (added 16/03/11) Letellier's paper is now available at: http://arxiv.org/abs/1103.2759

Answer 2

Both Tamas and Victor have pointed to the best current literature for general linear groups or others of Lie type, but maybe it's useful to add a series of small comments to supplement their answers.

1) The finite general linear groups, like their classical counterparts, are by far the best-behaved groups of Lie type for representation theory (both ordinary and modular) and related combinatorics. Even so, all aspects of the theory involve deep ideas and indirect approaches as Lusztig's work over many decades demonstrates. Moreover, even related groups like $GL, PGL, SL, PSL$ have character theories with varying degrees of difficulty. While Green's 1955 paper provides an algorithmic way to work out characters of finite general linear groups, the special linear groups have required much more sophisticated methods developed first by Deligne-Lusztig. It's usually not easy for groups of Lie type so say clearly what it means to "know" the characters of the group.

2) For any finite group, tensor products of irreducible representations (or products of characters) can be arbitrarily difficult to work out in detail even after character tables are in hand. It's much harder for groups of Lie type, where knowledge of characters is usually algorithmic. The best hope of getting uniform answers is to ask about tensor products of "generic" irreducibles, which predominate when both the underlying prime $p$ and its power $q$ grow large. Deligne-Lusztig theory constructs virtual characters, but "most" of these tend to be genuine characters and already show nice patterns in their distribution into series correlated with the sizes of classes in the Weyl group. Thus for $PGL_2(\mathbb{F}_q)$ roughly half of the characters have degree $q+1$ (principal series) and half have degree $q-1$ (discrete series). At the other extreme are the Steinberg character of degree $q$ and the trivial character (these are the "unipotent" characters).

3) As Victor points out, Lusztig's summary in rank 1 shows indirectly how to answer a natural qualitative question: in a "typical" tensor product of two irreducible characters, how often do principal series and discrete series characters appear? (Answer: about equally often.) Similar questions for other groups get much more complicated to settle.

4) Lusztig's character results for Lie types B, C, D show how dramatically the complexity increases, even though the results can be organized combinatorially. To study tensor products requires asking the right questions.

5) In the original question, Vinoth comments: but I'm not really interested in exceptional groups of Lie type (for those, this problem should be a standard mindless computation). Actually, to say anything interesting about tensor products here would require a creative approach. For example, what is shown by Lusztig about the characters of $E_8(q)$ is subtle and computationally not so easy to work with. There are 166 unipotent characters, whose degrees are polynomials in $q$, but roughly a third of them fail to occur as constituents of the character induced from the trivial character of a Borel subgroup. Knowing these characters is essential to producing the full character table, etc.

6) Since the Steinberg character occurs at the other extreme from "generic" characters, it's interesting to ask how its product with other irreducible characters will decompose. This has echoes in modular representation theory, as seen in recent work by Hiss and Zalesski, and is difficult to sort out even for type A. Lots of questions out there about tensoring.

Answer 3

You might be interested in Lusztig's paper arxiv:0805.0787 Tensor products of representations are closely related with tensor products of character sheaves; however character sheaves are not closed under tensoring. In the paper above Lusztig explains that one can slightly enlarge class of character sheaves to make it closed under tensoring. At least in a case of PGL(2) this is equivalent to the existence of very simple and nice patterns in decomposition of tensor products.